Double transgressions and Bott-Chern duality

We present a general framework for obtaining currential double transgression formulas on complex manifolds which can be seen as manifestations of Bott-Chern Duality. These results complement on one hand the simple transgression formulas obtained by Harvey -Lawson and on the other hand the double transgression formulas of Bismut-Gillet-Soulé. Among the applications we mention a Gysin isomorphism for Bott-Chern cohomology, an abstract Poincaré-Lelong formula for sections of holomorphic and Hermitian vector bundles implying Andersson's generalization of the standard Poincaré-Lelong, a Bott-Chern duality formula for the Chern-Fulton classes of singular varieties or a refinement of the first author's simple transgression formula for the Chern character of a Quillen superconnection associated to a self-adjoint, odd endomorphism. The existence of a Bismut-Gillet-Soulé double transgression without the hypothesis of degeneration along a submanifold stands out and is based on an extension to linear correspondences of the operation of morphism addition. Finally, as a by-product we also obtain a statement about the {pointwise localization} of the Samuel multiplicity of an analytic subvariety of a complex manifold along an irreducible component.

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